A343234 Triangle T read by rows: lower triangular Riordan matrix of the Toeplitz type with first column A067687.

1, 1, 1, 2, 1, 1, 5, 2, 1, 1, 12, 5, 2, 1, 1, 29, 12, 5, 2, 1, 1, 69, 29, 12, 5, 2, 1, 1, 165, 69, 29, 12, 5, 2, 1, 1, 393, 165, 69, 29, 12, 5, 2, 1, 1, 937, 393, 165, 69, 29, 12, 5, 2, 1, 1, 2233, 937, 393, 165, 69, 29, 12, 5, 2, 1, 1

Offset: 0

Author

Wolfdieter Lang, Apr 16 2021

Keywords

• nonn
• easy
• tabl

Comments

This infinite lower triangular Riordan matrix T is the so-called L-eigen-matrix of the infinite lower triangular Riordan matrix A027293 (but with offset 0 for rows and columns). Such eigentriangles have been considered by Paul Barry in the paper given as a link in A186020.

This means that E is the L-eigen-matrix of an infinite lower triangular matrix M if M*E = L*(E - I), with the unit matrix I and the matrix L with elements L(i, j) = delta_{i, j-1} (Kronecker's delta-symbol; first upper diagonal with 1's).

Therefore, this notion is analogous to calling sequence S an L-eigen-sequence of matrix M if M*vec(S) = L.vec(S) (or vec(S) is an eigensequence of M - L with eigenvalue 0), used by Bernstein and Sloane, see the links in A155002.

L*(E - I) is the E matrix after elimination of the main diagonal and then the first row, and starting with offset 0. Because for infinite lower triangular matrices L^{tr}.L = I (tr stands for transposed), this leads to M = L*(I - E^{-1}) or E = (I - L^{tr}*M)^{-1}.

Note that Gary W. Adamson uses a different notion: E is the eigentriangle of a triangle T if the columns of E are the columns j of T multiplied by the sequence elements B_j of B with o.g.f. x/(1 - x*G(x)), with the o.g.f. G(x) of column no. 1 of T. Or E(i, j) = T(i, j)*B(j). In short: sequence B is the L-eigen-sequence of the infinite lower triangular matrix T (but with offset 1): T*vec(B) = L.vec(B). See, e.g., A143866.

Thanks to Gary W. Adamson for motivating my occupation with such eigentriangles and eigensequences.

The first column of the present triangle T is A067687, which is then shifted downwards (Riordan of Toeplitz type).

Examples

			The triangle T begins:
n \ m   0   1   2   3  4  5  6  7  8  9 ...
-----------------------------------------
0:      1
1:      1   1
2:      2   1   1
3:      5   2   1   1
4:     12   5   2   1  1
5:     29  12   5   2  1  1
6:     69  29  12   5  2  1  1
7:    165  69  29  12  5  2  1  1
8:    393 165  69  29 12  5  2  1  1
9:    937 393 165  69 29 12  5  2  1  1
...
		

Crossrefs

Cf. A000041, A027293, A067687, A143866, A155002, A186020.

Formula

Matrix elements: T(n, m) = A067687(n-m), for n >= m >= 0, and 0 otherwise.

O.g.f. of row polynomials R(n,x) = Sum_{m=0..n} T(n, m)*x^m is

G(z, x) = 1/((1 - z*P(z))*(1 - x*z)), with the o.g.f. P of A000041 (number of partitions).

O.g.f. column m: G_m(x) = x^m/(1 - x*P(x)), for m >= 0.